Transcript Slide 1

A G2 and shape preserving
subdivision scheme for curve
interpolation
Chongyang Deng
2007-05-16
Outline
1.
2.
3.
4.
5.
6.
Introduction
The subdivision scheme
Example
Smoothness analysis
Generating spiral by subdivision scheme
Future work
Reference: Deng et al. A G2 and shape
preserving subdivision scheme for curve
interpolation. Submitted.
1.Introduction

Subdivision curve and surface
Main advantages:
1. Arbitrary topology
2. Efficiency
3. Simplicity …
1.Introduction

Classification of subdivision
1. Interpolation VS approximation
2. Linear VS nonlinear
1.Introduction




Linear schemes
Four point subdivision scheme
and its extensions
Dyn, N., Levin, D., and Gregory, J.A., 1987. A 4point interpolatory subdivision scheme for curve
design. CAGD, 4, 257-268.
Hassan, M.F., Ivrissimitzis, I.P., Dodgson, N.A.,
and Sabin, M.A., 2002. An interpolating 4-point
C2 ternary stationary subdivision scheme. CAGD,
19(1), 1-18.
1.Introduction

 

1

p    ω  pik  pik1  ω pik1  pik 2
2

 pik  pik1 pik1  pik 2 
pik  pik1


 2ω

2
2
2


k 1
2i  1
1
0ω
8
pik
k
i 1
k 1
2i 1
p
pik  pik1
2
p
k
i 1
p
p
2
k
i2
k
i 1
p
k
i 2
p
1.Introdution

Advantages of linear subdivision
schemes
1.Simple to implement
2.Easy to analyze
3.Affine invariance
1.Introdution

Disadvantages of linear subdivision
schemes:
Difficult to control the shape of
the limit curve (artifacts and
undesired inflexions)
1.Introdution

Example
1.Introdution

Example
1.Introdution

Example
1.Introdution
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
Nonlinear (geometric driven)
subdivision schemes
Yang Xunnian, Normal based
subdivision scheme for curve
design. CAGD 2006(23):243-260.
1.Introdution
k 1
2i 1
p
 (1  s ) p  s p  v
k
i
k
i 1
k
i
k
i
k
i
1.Introdution

Examples
2.The subdivision scheme
Outline
2.1 Origin idea
2.2 Preprocess
2.3 Adding new points
2.4 Calculating tangent vectors

2.The subdivision scheme
Origin idea:
C0: Adjacent two points run to equality
C1: Adjacent three points run to collinear
C2: Adjacent four points run to lie on a
circle

2.The subdivision scheme

Differential geometry:
For planar G2 continuous curve, the
tangent line and the osculating circle
(circle of curvature) at one point are the
first and second order approximants of
the curve near this point
2.The subdivision scheme

But it is complex to directly calculate
and compare the radii of the circles
passing three adjacent vertices!
So for each subdivision step, we select
the added points as like there is a G1
continuous circular arc spline
interpolating the vertices.
2.The subdivision scheme

Definition 1
(a) Convex edge (b) Inflexion edge (c) Straight edge
2.The subdivision scheme

Preprocess
2.The subdivision scheme

Adding new points
2.The subdivision scheme

The interpolating G1 arc spline
U
k 1
2i
 Ti
k
U
k 1
2 i 1
k
i 1
k
i 1
p

p
p
p
k
i
k
i
2.The subdivision scheme

The interpolating G1 arc spline
2ki1  2ki 1  0.5ik
2ki 11  2ki 11  0.5ik
2.The subdivision scheme

Calculating tangent vectors
   ik  k r k  r k
1
i 1
i
i 1


 ik    ik   ik1 ri k  ri k1
k




 2 i
k
i
k
k
i
k
  i 1  0
  i 1  0
ri 
k
pik pik1
2 sin  ik
2.The subdivision scheme

Why?
   ik  k r k  r k
1
i 1
i
i 1


 ik    ik   ik1 ri k  ri k1
k

  2  i

1
is used to control the
convergence rate.
k
i
k
k
i
k
  i 1  0
  i 1  0
2.The subdivision scheme
Inserting a line segment
 Analyze the shape of inflexion
1. The curvature is zero
2.The limit tangent vector can be
computed explicitly.

2.The subdivision scheme


Inserting a line segment
By picking the appropriate initial
tangent vector of two ends of a
edge we can insert a line segment
into the limit curve with G2
continuous.
2.The subdivision scheme

Inserting a line segment
3.Examples
1  0.5
1  0.25
3.Examples
Ternary four point subdivision scheme
3.Examples
1   2  0.5
1  0.5,  2  0.25
3.Examples
1  0.5,  2  0.25
3.Examples
1   2  0.5
1  0.5,  2  0.25
3.Examples
1  0.5,  2  0.25
4.Smoothness analysis




There step:
1.the polygon series converge.
2.the limit curve is G1 continuous.
3.the limit curve is G2 continuous.
4.Smoothness analysis

Convergence rate
5.Generate spiral by
subdivision scheme

Spirals are curves of one-signed,
monotone increasing or
decreasing curvature. They are
commonly perceived as high
quality profiles.
5.Generate spiral by
subdivision scheme

Aim: Generating spiral which
interpolating the given two
points and their tangent
vectors(G1 Hermite data).
5.Generate spiral by
subdivision scheme

Calculating tangent vectors
5.Generate spiral by
subdivision scheme

Examples
5.Generate spiral by
subdivision scheme

Examples
5.Generate spiral by
subdivision scheme

Examples
6.Future work


1. Matching admissable G2
Hermite data.
2. Interpolating point array
by spiral segments.
The end.
Thank you.